The motivic Newton–Hodge inequality and ordinariness conjectures

Let $X$ be a projective variety of dimension $d$ over a number field $F$, let $j:U\hookrightarrow X$ be the inclusion of a smooth dense open subset, and let $\pi:\mathscr{Y}\to U$ be a projective smooth scheme. For each integer $i$ and prime $\ell$, set

$$\mathscr{L}^i_{\ell}=R^i\pi_*\mathbb{Q}_{\ell},\qquad \overline{\mathscr{L}}^i_{\ell}=j_{!*}(\mathscr{L}^i_{\ell}[d])[-d].$$

Assume there is an André motive $M=M^{k,i}$ with $M_{\ell}\simeq H^k(X\otimes_FF^s,\overline{\mathscr{L}}^i_{\ell})$ for every $\ell$. Let $e$ be an idempotent endomorphism of $M$ in the category of André motives with $\mathbb{Q}$-coefficients, let $R$ be the direct summand cut out by $e$, and let $R_{\ell}$ be its $\ell$-adic étale realization.

The motivic Newton–Hodge inequality and ordinariness conjectures. For every integer $k$: (b) the representations $R_{\ell}$ form a strictly compatible system; (c) there exists a finite set $S=S(\pi,i,k,e)$ of primes of $F$ such that, for every prime $\ell$ and $\mathfrak{p}$ outside $S$ with $\mathfrak{p}\nmid\ell$,

$$\operatorname{NP}(\operatorname{Frob}_{\mathfrak{p}},R_{\ell})\ge\operatorname{HTP}(R_{\ell});$$

and (d) for infinitely many primes $\mathfrak{p}$ of $F$ and every prime number $\ell$,

$$\operatorname{NP}(\operatorname{Frob}_{\mathfrak{p}},R_{\ell})=\operatorname{HTP}(R_{\ell}).$$

These assertions are formulated after assuming the existence of the André motive in (a'). The source notes that (a') appears to follow from constructions of pure Nori motives by Ivorra and Morel, and reports verification of the polygon assertions for motives whose $\ell$-adic realizations become abelian on a finite-index subgroup; the general assertions remain open.